Swan constructed counterexamples over the rational numbers
E171228
Swan constructed counterexamples over the rational numbers refers to Richard G. Swan’s landmark result showing that certain invariant fields under finite group actions over the rational numbers are not rational, thereby disproving a general affirmative answer to Noether’s problem in this setting.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Swan constructed counterexamples over the rational numbers canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T1489782 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Swan constructed counterexamples over the rational numbers Context triple: [Noether's problem, knownResult, Swan constructed counterexamples over the rational numbers]
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A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
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B.
Kronecker’s finitism
Kronecker’s finitism is a philosophical and mathematical stance asserting that only finite, constructible mathematical objects and proofs are legitimate, rejecting the existence of actual infinities.
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C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
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D.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
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E.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Swan constructed counterexamples over the rational numbers Target entity description: Swan constructed counterexamples over the rational numbers refers to Richard G. Swan’s landmark result showing that certain invariant fields under finite group actions over the rational numbers are not rational, thereby disproving a general affirmative answer to Noether’s problem in this setting.
-
A.
Hilbert’s irreducibility theorem
Hilbert’s irreducibility theorem is a fundamental result in number theory and algebraic geometry that ensures many polynomial equations with parameterized coefficients retain irreducibility for infinitely many specializations of those parameters.
-
B.
Kronecker’s finitism
Kronecker’s finitism is a philosophical and mathematical stance asserting that only finite, constructible mathematical objects and proofs are legitimate, rejecting the existence of actual infinities.
-
C.
Hilbert’s Nullstellensatz
Hilbert’s Nullstellensatz is a foundational theorem in algebraic geometry that establishes a deep correspondence between ideals in polynomial rings and algebraic sets, linking algebra and geometry.
-
D.
Israel–Carter–Robinson uniqueness theorems
The Israel–Carter–Robinson uniqueness theorems are a set of results in general relativity showing that stationary, asymptotically flat black holes in four-dimensional spacetime are completely characterized by just their mass, charge, and angular momentum.
-
E.
Surreal numbers
Surreal numbers are a class of numbers introduced by John H. Conway that form an extensive ordered field encompassing the real numbers, infinite quantities, and infinitesimals within a unified framework.
- F. None of above. chosen
Statements (36)
| Predicate | Object |
|---|---|
| instanceOf |
counterexample to Noether's problem
ⓘ
mathematical result ⓘ |
| author | Richard G. Swan NERFINISHED ⓘ |
| concerns |
fields of invariants under finite group actions
ⓘ
finite group actions on rational function fields ⓘ |
| context |
Galois theory
ⓘ
algebraic geometry ⓘ field theory ⓘ invariant theory ⓘ |
| disproves | general affirmative answer to Noether's problem over Q ⓘ |
| field | field of rational numbers ⓘ |
| impact |
changed understanding of rationality questions in invariant theory
ⓘ
motivated further study of obstructions to rationality ⓘ |
| influenced |
research on rationality of quotient varieties
ⓘ
subsequent work on Noether's problem ⓘ |
| involves |
finite groups with nonrational invariant fields over Q
ⓘ
nonrational invariant fields ⓘ |
| mainSubject |
Noether's problem
ⓘ
rationality problem for fields of invariants ⓘ |
| mathematicsSubjectClassification |
11R32
ⓘ
13A50 ⓘ 14E08 ⓘ |
| namedAfter | Richard G. Swan NERFINISHED ⓘ |
| overField |
Q
ⓘ
rational numbers ⓘ |
| relatedTo |
Emmy Noether
ⓘ
Noether's problem ⓘ
surface form:
Noether's problem for finite groups
algebraic tori ⓘ rationality of fields of invariants ⓘ unramified Brauer group ⓘ |
| shows |
Noether's problem has a negative answer over the rational numbers for some finite groups
ⓘ
existence of nonrational invariant fields over the rationals ⓘ not all fields of invariants of finite group actions on rational function fields over Q are purely transcendental ⓘ |
| timePeriod | 20th century ⓘ |
| usedMethod |
Brauer group obstructions
ⓘ
cohomological techniques ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Swan constructed counterexamples over the rational numbers Description of subject: Swan constructed counterexamples over the rational numbers refers to Richard G. Swan’s landmark result showing that certain invariant fields under finite group actions over the rational numbers are not rational, thereby disproving a general affirmative answer to Noether’s problem in this setting.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.